Limit Experiments of GARCH

GARCH is one of the most prominent nonlinear time series models, both widely applied and thoroughly studied. Recently, it has been shown that the COGARCH model, which has been introduced a few years ago by Klüppelberg, Lindner and Maller, and Nelson’s diffusion limit are the only functional continuous-time limits of GARCH in distribution. In contrast to Nelson’s diffusion limit, COGARCH reproduces most of the stylized facts of financial time series. Since it has been proved, that Nelson’s diffusion is not asymptotically equivalent to GARCH in deficiency, we investigate in the present paper the relation between GARCH and COGARCH in Le Cam’s framework of statistical equivalence. We show that GARCH converges generically to COGARCH, even in deficiency, provided that the volatility processes are observed. Hence, from a theoretical point of view, COGARCH can indeed be considered as a continuous-time equivalent to GARCH. Otherwise, when the observations are incomplete, GARCH still has a limiting experiment which we call MCOGARCH, and which is not equivalent, but nevertheless quite similar to COGARCH. In the COGARCH model, the jump times can be more random, as for the MCOGARCH, a fact practitioners may see as an advantage of COGARCH. AMS 2000 subject classifications: Primary 62B15, 60J25; secondary 91B84, 91B70